APSTA-GE 2003: Intermediate Quantitative Methods

class: center, middle, inverse, title-slide

# APSTA-GE 2003: Intermediate Quantitative Methods
## Lab Section 003, Week 4
### New York University
### 09/29/2020

---

## Reminders

- Assignment 2
  - Due: **10/02/2020 11:55pm (EST)**
- Office hours
  - Monday 9 - 10am (EST)
  - Wednesday 12:30 - 1:30pm (EST)
- Office hour Zoom link
  - https://nyu.zoom.us/j/97347070628 (pin: 2003)
- Office hour notes
  - Available on NYU Classes under the "Resources" tab
  
---
  
## Fun slide

- **"What do you call a regression model with only two data points?"**

- A line.

---

## Let's build the fun part

**Let's create a simulated data set with only two data points.**

```r
first_point <- c(25, 10)
second_point <- c(5, 30)
dat_for_fun <- data.frame(
 X = c(first_point[1], second_point[1]),
 Y = c(first_point[2], second_point[2])
)
dat_for_fun
```

```
##    X  Y
## 1 25 10
## 2  5 30
```

---

## Here comes the fun part

**Let's fit a regression model and talk about the perfect R-squared and those NAs for half an hour.**

```r
# Let's fit a linear regression model
lm_for_fun <- lm(formula = Y ~ X, data = dat_for_fun)
summary(lm_for_fun)
```

```
## 
## Call:
## lm(formula = Y ~ X, data = dat_for_fun)
## 
## Residuals:
## ALL 2 residuals are 0: no residual degrees of freedom!
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)       35         NA      NA       NA
## X                 -1         NA      NA       NA
## 
## Residual standard error: NaN on 0 degrees of freedom
## Multiple R-squared:      1,	Adjusted R-squared:    NaN 
## F-statistic:   NaN on 1 and 0 DF,  p-value: NA
```

---

## More fun? Yes!

**To make things even better, let's plot the results!**

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---

## How about this?

**How about four data points?**

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---

## Let's check the result

*(25, 10), (10, 15), (5, 30), (30, 20)*

```r
lm2_for_fun <- lm(formula = Y ~ X, data = dat2_for_fun)
summary(lm2_for_fun)
```

```
## 
## Call:
## lm(formula = Y ~ X, data = dat2_for_fun)
## 
## Residuals:
##      1      2      3      4 
## -5.882  6.471  6.029 -6.618 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)  25.4412     8.7181   2.918    0.100
## X            -0.3824     0.4293  -0.891    0.467
## 
## Residual standard error: 8.849 on 2 degrees of freedom
## Multiple R-squared:  0.284,	Adjusted R-squared:  -0.07395 
## F-statistic: 0.7934 on 1 and 2 DF,  p-value: 0.4671
```

---

## Seamlessly switching to today's topics

**Distance to the mean**: the distance between each data point and the average DV value.

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---

## Sum of Squares (SS)

What is **Sum of Squares**?

The sum of squared distance.

`$$\sum_{i=1}^n \ \ \ (Y_i - \overline{Y}) \ \ \ ^2$$`

---

## Three types of SS

**Total Sum of Squares (TSS)**: Sum of squared distance between observed data and the mean

`$$TSS = \sum_{i=1}^n (Y_i - \overline{Y})^2$$`

**Model (Explained) Sum of Squares (MSS)**: Sum of squared distance included in the model

`$$MSS = \sum_{i=1}^n (\hat{Y_i} - \overline{Y_i})^2$$`

**Residual Sum of Squares (RSS)**: Sum of squared distance not included in the model

`$$RSS = \sum_{i=1}^n e_{i}^2 = \sum_{i=1}^n (Y_i - \hat{Y_i})^2$$`

---

## Total Sum of Squares (TSS)

`$$TSS = \sum_{i=1}^n (Y_i - \overline{Y})^2$$`

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---

## Total Sum of Squares (TSS) cont.

`$$\begin{align}
TSS_{fun} & = \sum_{i=1}^4 (Y_i - \overline{Y})^2 \\
          & = (10 - 18.75)^2 + (15 - 18.75)^2 + (30 - 18.75)^2 + (20 - 18.75)^2 \\
          & = 218.75
\end{align}$$`

---

## Model Sum of Squares (MSS)

`$$MSS = \sum_{i=1}^n (\hat{Y_i} - \overline{Y_i})^2$$`

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---

## Model Sum of Squares (MSS) cont.

`$$\begin{align}
MSS_{fun} & = \sum_{i=1}^4 (\hat{Y_i} - \overline{Y_i})^2 \\
          & = (15.88235 - 18.75)^2 + (23.52941 - 18.75)^2 + (13.97059 - 18.75)^2 + (21.61765 - 18.75)^2 \\
          & = 62.1323529412
\end{align}$$`

---

## Residual Sum of Squares (RSS)

`$$RSS = \sum_{i=1}^n e_{i}^2 = \sum_{i=1}^n (Y_i - \hat{Y_i})^2$$`

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---

## Residual Sum of Squares (RSS) cont.

`$$\begin{align}
RSS_{fun} & = \sum_{i=1}^4 (Y_i - \hat{Y_i})^2 \\
          & = (10 - 15.88235)^2 + (30 - 23.52941)^2 + (20 - 13.97059)^2 + (15 - 21.617656)^2 \\
          & = 156.617732353
\end{align}$$`

---

## TSS, MSS, and RSS

In our case:

`$$TSS_{fun} = 218.75 \\
MSS_{fun} = 62.1323529412 \\
RSS_{fun} = 156.617732353$$`

`$$TSS_{fun} = 218.75\\
MSS_{fun} + RSS_{fun} = 62.1323529412 + 156.617732353 = 218.750085294 \\ 
TSS_{fun} = MSS_{fun} + RSS_{fun}$$`

---

## Analysis of Variance

```r
anova(lm2_for_fun)
```

```
## Analysis of Variance Table
## 
## Response: Y
##           Df  Sum Sq Mean Sq F value Pr(>F)
## X          1  62.132  62.132  0.7934 0.4671
## Residuals  2 156.618  78.309
```

`$$F_{fun} = \frac{MSS_{fun}/(p-1)}{RSS_{fun}/(n-p)} = \frac{62.132}{156.618/(4 - 2)} = 0.793$$`

where `$p$` is the degrees of freedom.

---

## R Squared

`$R^2$` is the percentage of variability in the data set explained by the model.

`$$R^2 = \frac{MSS}{TSS} = 1 - \frac{RSS}{TSS}$$`

In our case:

`$$\begin{align}
R_{fun}^2 & = \frac{MSS_{fun}}{TSS_{fun}} = 1 - \frac{RSS_{fun}}{TSS_{fun}} \\
          & = \frac{62.132}{218.75} = 1 - \frac{156.618}{218.75} \\
          & = 0.284
\end{align}$$`